In this paper, we introduce and investigate a new regularity condition in the asymptotic sense for optimization problems whose objective functions are polynomial. The normalization argument in asymptotic analysis enables us to study the existence as well as the stability of solutions to these problems. We prove a Frank-Wolfe type theorem for regular optimization problems and an Eaves type theorem for non-regular pseudoconvex optimization problems. Moreover, under the regularity condition, we show results on the stability such as upper semicontinuity and local upper-H\"{o}lder stability of the solution map of polynomial optimization problems. At the end of the paper, we discuss the genericity of the regularity condition.